Sunday, 13 April 2014

Secular Stagnation and Three Period OLG

For macroeconomists. This
post is a kind of introduction to the new paper on secular stagnation by Eggertsson and
Mehrotra. As usual, any misinterpretations are my fault.

A basic idea behind secular stagnation is that the natural real
rate of interest might become negative for a prolonged period of time. A simple
way to model this would be to allow the steady state real interest rate to
become negative. That cannot happen in basic representative agent models, where
the steady state real interest rate (absent growth) is given by

1+r = 1/b

where b<1
is the utility discount factor. With population growth (at rate = n) this
becomes

1+r = n +1/b

Note that a fall in n will reduce the real interest rate, which
is a useful result if we want to relate secular stagnation to falling
population growth, but rates cannot fall below the rate of time preference.

In a standard two period OLG model we have more flexibility. If
agents only work in the first period, then they need to save in that period to
be able to smooth consumption between their working lives and retirement. If we
allow them to do that through investing in capital, and if α is the exponent on
capital in a Cobb Douglas production function, then with log utility the real
interest rate in steady state is given by

r = k + kn where k
= α(1+b)/b(1- α)

If one period is about 25 years, then b could be 0.5 (annual b = 0.973),
and with α = 0.4 then k=2. So now the impact of a fall on population growth on
the real interest rate is magnified, but the steady state real interest rate is
also likely to be above the representative agent case. (If n=0 and b =
0.5, then we have r=1 and r=2 respectively. For a 25 year period this would
correspond to annual interest rates of around 2.8% and 4.5%.)

In a three period OLG setup, we can have saving without
capital. The middle aged work (receiving income Y), and they lend to the young,
and in retirement get paid back by the now middle aged. Suppose, however, that because
of some credit friction the amount the young can borrow gross of interest payments is fixed at D, and let d=D/Y<1. The
middle aged would like to lend them enough to smooth consumption, so the supply
of loans in steady state is (given log utility)

b (Y-D)/
(1+b)

where Y-D is middle age income net of repaying loans taken out
when young. The demand for loans is

D (1+n)/(1+r)

The borrowing limit is gross of interest, so with no population
growth actual borrowing is D/(1+r). With population growth there are more of
the young than middle aged, so we need to scale up loan demand accordingly. The
real interest rate equates demand and supply, which implies

1+r = j + jn where j
= (1+b)d/b(1-d)

Now if d is small, j could be less than one, which reduces the
sensitivity of interest rates to population growth, although a fall in population growth still reduces rates. However this also means
that the gross interest rate (1+r) could be less than one, so the steady state
real interest rate could be negative.

The middle aged need to save for retirement, but the only way
they can do this is by lending to the young. The higher the real interest rate,
the less the young can borrow because of the credit friction. In that situation,
the real interest rate could easily be negative, because only then will the
young be able to borrow enough to allow the middle age to consumption smooth
when they retire.

The key result that Eggertsson and Mehrotra explore is that a
credit crunch - a fall in D - could lower real interest rates into negative
territory, and could therefore generate secular stagnation. They consider how
inequality could be incorporated into the model, and then embed the model in a
nominal framework. Nominal wage rigidity is added (using a similar mechanism to
that in the Schmitt-Grohe and Uribe paper I discussed here), and the implications for monetary and
fiscal policy explored. So I have only touched on the paper here, but as this three period OLG set-up is not standard I thought this post might be
useful.

4 comments:

"The higher the real interest rate, the less the young can borrow because of the credit friction."

I'm glad you pointed this out, because in this model cutting interest rates raises consumption for no other reason than that, for an assumed cap on principal plus interest, lenders can then lend a higher principal (apart from a temporary transitional effect when rates change). As a role for the interest rate, this is rather unconventional and they perhaps should have said more about it. It's also worth noting that if you instead assume that the credit cap applies to principal net of interest, that the relationship between interest rates and income is reversed.

Nick Rowehttp://worthwhile.typepad.com/worthwhile_canadian_initi/2014/04/on-forgetting-land-in-models-of-secular-stagnation.htmlcriticises this paper because the existence of land (or any other asset which has a real positive rental yield) blows up the model. The middle aged don't bother lending to the young at negative real rates, they buy land off the old, keep the rents, and then sell the land to the next lot of middle aged. I'm not an economist, so can't tell who is right here, although Prof Rowe does seem to make sense. Another good argument for a land value tax?

That is what is traditionally going on in two period OLGs: normally with capital, but land or government debt will do. In the real world that obviously happens. However its not clear to me why the real return on land has to be positive, as Nick asserts.

It is also true that the young borrow. So we need a model with both, which is what the authors of this paper say they intend to do next. I look forward to seeing whether this 'blows up' their analysis.